Game Theoretic Analysis, pp. 203-221 (2019)No Access
Chapter 10: Cournot Equilibrium Uniqueness: At 0 Discontinuous Industry Revenue and Decreasing Price Flexibility
Pierre Von Mouche and Takashi Sato
Wissenschaftliche Publikationen von Pierre von Mouche.
Publikationen von P. v. Mouche.
Die Publikationsliste ist in umgekehrter chronologischer Reihenfolge.
2021 (mit W. Heijman).
Media Bias: A Game Theoretic Approach. Zur Publikation angeboten.
2021 (mit J.-i. Itaya).
Equilibrium Uniqueness in Aggregative Games: Very Practical Conditions. Zur Publikation angeboten.
Sufficient conditions for a broad class of aggregative games with
strictly quasi-concave conditional payoff functions to have a unique Nash equilibrium are presented.
The setting presupposes that each player has ℝ+
as strategy set, makes smoothness assumptions but allows for a discontinuity of stand-alone payoff functions at 0.
The conditions are completely in terms
of marginal reductions which may be considered as
primitives of the game. For the bulk of games in the literature they easily
can be checked.
The results are proved by means of the Selten-Szidarovszky technique and are illustrated by reproducing
quickly a highly non-trivial result from the literature.
2021 (mit T. Iimura).
Discrete Hotelling Pure Location Games: Potentials and Equilibria.
Zur Publikation angeboten.
We study two-player one-dimensional discrete Hotelling pure location games
assuming that demand f(d) as a function of distance d is constant or strictly decreasing.
We show that this game admits a best-reply potential. This result holds in particular for
f(d) = wd with 0 < w ≤ 1. For this case special attention will be given to the structure of the equilibrium set and a conjecture about the increasingness
of best reply correspondences will be made.
The Continuous Hotelling Pure Location Game with Elastic Demand Revisited.
In: MOTOR 2020, LNCS 12095, 246--262. Editors: A. Kononov et al.
Springer Nature Switzerland AG.
The Hotelling pure location game is revisited.
It is assumed that there there are two identical players, strategy sets are one-dimensional
and demand as a function of distance is constant or strictly decreasing.
Besides qualitative properties of conditional payoff functions,
attention is given to the structure of the equilibrium set, best-response
correspondences and existence of potentials.
2020 (mit R. Haagsma).
Securing Land Rights under Rapid Population Growth: the Feasibility of Institutional
Land Rights Protection in Africa.
Journal of Institutional and Theoretical Economics 176, 2, 312--350.
The paper examines the claim that a virtuous cycle of more secure land rights,
more land-saving investments, and denser populations requires the development
of institutions that regulate competition over land. We construct a contest model that links
the tenure security-investment relationship to the efforts of land users to enhance land rights
themselves and the role of institutional protection. We study the impact of population growth
on a close-to-subsistence economy, including the possibility that it weakens institutional protection.
We derive sufficient conditions for a positive effect on land investment,
but also show that population growth can push the economy into a low-productivity trap.
2020 (mit A. Hagen und H.-P. Weikard).
The Two-Stage Game Approach to Coalition Formation: Where we Stand and Ways to Go.
Games, 11(1), 3.
Coalition formation is often analysed
in an almost non-cooperative way as a two-stage game that consists of a first stage comprising membership actions and a second stage with physical actions, such as the provision of a public good. We formalise this widely used approach for the case where in each stage actions are simultaneous.
There is special attention to the case of a symmetric physical game.
Various theoretical results, in particular for cartel games, are provided.
As they are crucial, also recent results on uniqueness of coalitional
equilibria of Cournot-like physical games are reconsidered.
Various concrete examples are included.
Finally, we discuss research strategies
to obtain results about equilibrium coalition structures
with abstract physical games in terms of qualitative properties of their primitives.
2019 (mit T. Sato).
Cournot Equilibrium Uniqueness: at 0 Discontinuous
Industry Revenue and Decreasing Price Flexibility.
International Game Theory Review, 21, 2, 1--19.
We consider the equilibrium uniqueness problem for
a large class of Cournot oligopolies with convex cost functions
and a proper price function p̃ with
decreasing price flexibility. This class allows for (at 0) discontinuous
industry revenue and in particular for p̃(y) = y- α.
The paper illustrates in an exemplary way
the Selten-Szidarovszky technique based
on virtual backward reply cspondences. An algorithm for the
calculation of the unique equilibrium is provided.
2018 (mit T. Iimura und T. Watanabe).
Binary Action Games: Deviation Properties,
Semi-Strict Equilibria and Potentials. Discrete Applied Mathematics, 251, 57--68.
For binary action games we present three properties which have in common that they are defined by conditions on marginal payoffs.
The first two properties guarantee the existence of a special type of
Nash equilibrium called semi-strict Nash equilibrium,
for which we also show an algorithm to locate.
The third one guarantees the existence of an exact potential,
and can realize the aforementioned two properties
in a class of exact potential games.
The first one guarantees the existence of a generalized ordinal potential.
Each symmetric binary action game possesses all the three properties.
The results are illustrated by three applications.
2018 (mit N. Kukushkin).
Cournot Tatonnement and Nash Equilibrium in Binary Status Games.
Economics Bulletin, 38, 2, 1038--1044.
We study a rather simplified game model of competition for status. Each player
chooses a scalar variable (say, the level of conspicuous consumption), and then those
who chose the highest level obtain the "high" status, while everybody else remains
with the "low" status. Each player strictly prefers the high status, but they also have
intrinsic preferences over their choices. The set of all feasible choices may be continuous
or discrete, whereas the strategy sets of different players can only differ in their upper
and lower bounds. The resulting strategic game with discontinuous utilities does not
satisfy the assumptions of any general theorem known as of today. Nonetheless, the
existence of a (pure strategy) Nash equilibrium, as well as the "finite best response
improvement property," are established.
(mit W. Pijnappel). The Hotelling Bi-matrix Game.
Optimization Letters, 12, 1, 187--202, Kopie;
We study the pure equilibrium set
for a specific symmetric finite game in strategic form, referred to as the Hotelling bi-matrix game.
General results that guarantee non-emptiness of this set (for all parametric values)
do not seem to exist. We prove its
non-emptiness by determining the pure equilibrium set.
In this proof so-called demi-modality properties of the conditional payoff functions play an important role.
2018 (mit F. Quartieri).
Cournot Equilibrium Uniqueness via Demi-Concavity.
Optimization, 67, 4, 441--455,
A family of oligopolies that possess a unique equilibrium was identified in
the second authors doctoral dissertation. For such a family, it is therein
specified a class of functions--economically related to the price function of a Cournot
oligopoly---that satisfy a particular type of quasi-concavity. The first
part of the present article (i) conceptualizes that type of quasi-concavity
by introducing the notion of demi-concavity, (ii) considers two possible
variants and (iii) provides some calculus properties. The second part, by
relying on the results on demi-concavity, proves a Cournot equilibrium
uniqueness theorem which is new for the journal literature and subsumes
various results thereof. A third part shows an example that illustrates the
novelty of the result.
2017 (mit T. Iimura und T. Watanabe).
Best-Response Potential for Hotelling Pure Location Games.
Economics Letters, Volume 160, 73--77,
We revisit two-person one-dimensional pure location games
à la Anderson et al. (1992)
and show that they admit continuous best-response potential functions (Voorneveld, 2000)
if demand is sufficiently elastic
(to the extent that
the Principle of Minimum Differentiation fails);
if demand is not that elastic
(or is completely inelastic)
they still admit
continuous quasi-potential functions (Schipper, 2004).
We also show that, even if
a continuous best-response potential function exists,
a generalized ordinal potential function
(Monderer and Shapley, 1996) need not exist.
2017 (mit T. Iimura und T. Watanabe).
Best-reply Potential for Two-Person One-Dimensional Pure Location Games.
Tokyo Metropolitan University Research Paper Series, No. 178.
We study two-person one-dimensional non-price pure location games à
la Anderson et al. (1992) under the setting that the strategy set is either a compact real interval or
a finite integer interval and demand as a function of distance is constant (inelastic) or
strictly decreasing (elastic). We show that on a finite integer interval, the game is a best-
reply potential game (Voorneveld, 2000); on a compact real interval, it is a best-reply
potential game if the demand is a sufficiently decreasing, strictly decreasing continuous
function of distance; otherwise a quasi-potential game (Schipper, 2004). We also show
that, even if a best-reply potential exists, a generalized ordinal potential (Monderer and
Shapley, 1996) need not exist. Thus, on a finite integer interval, the game generally
lacks the finite improvement property (Monderer and Shapley, 1996) but has the finite
best-reply property (Milchtaich, 1996); on a compact real interval, the existence of a
pure Nash equilibrium is secured by the existence of some continuous potential function,
which, as we shall show, is indeed the case.
2016 (mit R. Haagsma).
On the Endowment Effect in `Apple-Mars' Experiments.
Apstract, 10, 2-3, 47--50,
In this article we take a close look at a specific type of
behavioural experiment that Antonides conducted to study the endowment
effect. We argue that if such experiments ignore to test for the presence of
persons in the sample who are indifferent between alternatives, the
identification procedure for establishing an endowment effect is fallible.
2016. The Selten-Szidarovszky Technique: the Transformation Part.
In: Recent Advances in Game Theory and Applications, 147--164. Editors: L. Petrosyan and V. Mazalov. Birkhauser. ISBN 9783319438382.
A technique due to Selten and Szidarovszky for the analysis of Nash equilibria
of games with an aggregative structure
Among other things it is shown that the transformation part of
this technique can be extended to abstract games
with co-strategy mappings
and that this part allows for a purely algebraic setting.
2016 (mit F. Quartieri).
Equilibrium Theory for Cournot Oligopolies and
Related Games: Essays in Honour of Koji Okuguchi.
Springer Series in Game Theory. Editors: P. v. Mouche
and F. Quartieri. ISBN 978-3-319-29253-3.
This state-of-the-art collection of papers on the theory of Cournotian competition focuses on two main subjects: oligopolistic Cournot competition and contests. The contributors present various applications of the Cournotian Equilibrium Theory, addressing topics such as equilibrium existence and uniqueness, equilibrium structure, dynamic processes, coalitional behavior and welfare. Special emphasis is placed on the aggregative nature of the games that are relevant to such theory. This contributed volume was written to celebrate the 80th birthday of Prof. Koji Okuguchi, a pioneer in oligopoly theory.
2016. On the Geometric Structure of the Cournot Equilibrium Set:
the Case of Concave Industry Revenue and Convex Costs.
In: Equilibrium Theory for Cournot Oligopolies and
Related Games: Essays in Honour of Koji Okuguchi.
Springer Series in Game Theory, 63--88. Editors: P. v. Mouche and F. Quartieri. ISBN 978-3-319-29253-3.
The recent results
in  on equilibrium (semi-)-uniqueness for
homogeneous Cournot oligopolies with
concave industry revenue and convex costs are refined
and conceptualised. For this class of oligopolies also new results concerning
the geometric structure of the equilibrium set E are provided. In particular, a subclass is identified
for which E
is a non-empty polytope on which the equilibrium aggregator is
constant and a subclass for which E
is a 1-dimensional polytope on which the equilibrium aggregator
2015 (mit W. Heijman).
Floquet Theory and Economic Dynamics II.
WASS Working Paper No. 15. DOI:10.13140/RG.2.1.1067.4009.
[Verbesserung von: 1996
(mit W. Heijman).
Floquet Theory and Economic Dynamics (Extended Version).
Wageningen Economic Papers, 1996-5.
Und Verbesserung von: 1993 (mit W. Heijman).
Floquet Theory and Economic Dynamics.
Wageningen Economic Papers, 1993-3.]
Floquet theory is an appropriate tool
for studying ordinary linear recurrence and differential
equations with periodic coefficients, and is a generalization
of the theory for constant coefficients.
Floquet theory has still not found its way into economics, although
it seems to be relevant for economic dynamics.
As well as a discussion of this relevance and an illustration of it
in the context of the Samuelson-Hicks multiplier-accelerator model,
this article contains an appendix that provides a quite complete exposition
of Floquet theory for recurrence equations.
2015 (mit N. Abudaldah, W. Heijman und P. Heringa).
Return of the Icecream Men. A Discrete Hotelling Game.
Romanian Journal of Regional Science, 9, 2, 39-48.
[Verbesserung von: 2015
(mit W. Heijman, P. Heringa und N. Abudaldah).
Return of the Icecream Men. A Discrete Hotelling Game.
WASS Working Paper No. 11.
We consider a finite symmetric game in strategic form with
two players which can be interpreted as a discrete variant of the Hotelling game in a one or two-dimensional space.
As the analytical investigation of this game is tedious, we simulate with Maple and formulate some conjectures.
In addition we present a short literature overview.
2015 (mit H. Folmer).
Nash Equilibria of Transboundary Pollution Games.
In: Handbook of Research Methods and Applications in Environmental Studies, 504--524. Edward-Elgar. Editor: M. Ruth.
We reconsider the Nash equilibrium existence and uniqueness problem for transboundary pollution games.
There is special attention for the equilibrium set E for
effective compact transboundary pollution games with continuous strictly concave
production functions, continuous convex damage cost functions
and uniformly distributed transboundary pollution.
For this case we show that E is a non-empty polytope and that
for each country all equilibrium deposition levels are equal.
If in addition each damage cost function is differentiable, then there is a unique equilibrium.
The results are obtained by exploiting the aggregative structure of transboundary pollution games.
2015 (mit F. Quartieri).
Cournot Equilibrium Uniqueness in Case of Concave Industry Revenue: a Simple Proof.
Economics Bulletin, 35, 2, 1299-1305.
We provide a simple proof of an equilibrium uniqueness result by Murphy, Sherali and Soyster
for homogeneous Cournot oligopolies with concave industry revenue function and convex cost functions.
We show how to adapt this proof to obtain substantial improvements of this result concerning capacity constraints, non-differentiable cost functions and industry revenue functions that are discontinuous at 0.
2015 (mit T. Yamazaki).
Sufficient and Necessary Conditions for Equilibrium Uniqueness in Aggregative Games.
Journal of Nonlinear and Convex Analysis, 16, 2, 353-364.
We identify sufficient and necessary conditions for an aggregative game to have a unique Nash equilibrium.
In particular, an improvement of a result of Gaudet and Salant (1991) for
Cournot oligopolies is obtained. The results are obtained by exploiting the general relation between Nash equilibria
and fixed points of the (virtual) aggregate cumulative best reply correspondence.
2014 (mit M. Finus und B. Rundshagen).
On Uniqueness of Coalitional Equilibria.
In: Contributions to Game Theory and Management. Volume VII, 51-60.
Editors: L. Petrosjan, N. Zenkevich.
St. Petersburg State University. ISSN 2310-2608.
[Verbesserung von: 2005 (mit M. Finus und B. Rundshagen)
Uniqueness of Coalitional Equilibria.
Fondazione Eni Enrico Mattei, Nota de Lavaro 23.2005.
In the so-called `new approach' of coalition formation it is important
that coalitional equilibria are unique.
Uniqueness comes down to existence and to semi-uniqueness,
i.e. that there exists at most one equilibrium.
Although conditions for existence are not problematic, conditions for semi-uniqueness are.
We provide semi-uniqueness conditions by deriving a new equilibrium
semi-uniqueness result for games in strategic form
with higher dimensional strategy sets. The result applies in particular to
2014 (mit P. Bertoletti). Inferior Factor in Cournot Oligopoly Revisited.
Journal of Economics, 112, 85-90.
We reconsider the recent work by
Okuguchi (2010) on (possibly asymmetric)
Cournotian firms with two production factors, one being inferior for
each firm. It is shown there that an increase in the price of the inferior
factor does raise the equilibrium industry output. In addition of providing
a simpler and more rigorous proof of that result, we generalize it to the
case of technologies with s ≥ 2 factors and also allow some firms not to use the inferior one.
2013 (mit F. Quartieri).
On the Uniqueness of Cournot Equilibrium in Case of Concave Integrated Price Flexibility.
Journal of Global Optimization, 57, 3, 707-718.
We consider a class of homogeneous Cournot oligopolies with concave
integrated price flexibility and convex cost functions. We provide new
results about the semi-uniqueness and uniqueness of (Cournot) equilibria for
the oligopolies that satisfy these conditions. The condition of concave
integrated price flexibility is implied by (but does not imply) the
log-concavity of a continuous decreasing price function. So, our results
generalize previous results for decreasing log-concave price functions and
convex cost functions. We also discuss the particular type of
quasi-concavity that characterizes the conditional revenue and profit
functions of the firms in these oligopolies and we point out an error of the
literature on the equilibrium uniqueness in oligopolies with log-concave
price functions. Finally, we explain how the condition of concave
integrated price flexibility relates to other conditions on the price and
revenue functions usually considered in the literature, e.g., the concavity
of the price function or the concavity of the aggregate revenue associated
to a price function.
2013 (mit H. Folmer). Analysing the Folk Theorem for Linked Repeated Games.
In: Contributions to Game Theory and Management. Volume VI, 146-164.
Editors: L. Petrosjan, N. Zenkevich. Graduate School of Management
[Verbesserung von: 2007 (mit H. Folmer)
Linking of Repeated Games: When does it Lead to More Cooperation
and Pareto Improvements? Fondazione Eni Enrico Mattei, Nota de Lavaro 60.2007.
We deal with the linkage of infinitely repeated games.
Results are obtained by analysing the relations of the feasible individually rational payoff regions
of the isolated games and the linked game.
In fact we have to handle two geometric
problems related to Minkowski sums, intersections and Pareto boundaries of convex sets.
2013 (mit F. Quartieri und F. Szidarovszky). On a Fixed Point Problem Transformation Method.
Proceedings of the 10th international conference on fixed point theory and its applications (ICFPTA),
179--190. ISBN 978-606-17-0420-0. ISSN 1661-7738.
We show how the fixed point problem for a special type of correspondence R
which satisfies a factorisation property can be handled
by considering an associated more simple fixed point problem for a correspondence B
with domain typically a subset of ℝ.
In addition we analyse the fixed point problem for B
under additional conditions on R
that guarantee that B is at most singleton-valued.
In fact we generalize, improve and make more conceptual a game theoretic technique developed by Selten and Szidarovszky.
2013 (mit W. Heijman).
A Procedure for Determining an Optimal Landscape and its Monetary Value.
In: The Economic Value of Landscapes.
Routledge Studies in Ecological Economics, Volume 26, 123--135. Editors C. van der Heide and W. Heijman.
The article corrects, improves and formalises the procedure
in Heijman and Goossen (2009) to determine an optimal landscape and its monetary value,
assuming a Cobb-Douglas benefit function.
2013 (mit R. Haagsma). Egalitarian Norms, Economic Development, and Ethnic Polarization.
Journal of Comparative Economics, 41, 719-744.
Economic development generally implies that traditional
egalitarian norms and beliefs are replaced by modern individualistic values.
Particularly when opportunities for advancement are unequally presented to
people, this transformation may be accompanied by polarization and violent conflict. We
illustrate this point by describing the processes of land privatization in
Sub-Saharan Africa and then present two models that capture some salient
aspects of this transformation in rural communities, including the possibility of polarization. We find that the
support of egalitarian norms is notably strong when new opportunities are
available for only a few people or when the community is socially unstable.
Moreover, in unstable communities, polarization is strongest
when the group with the most lucrative opportunities comprises half the population.
2012 (mit F. Quartieri).
Existence of Equilibria in Cournotian Games with Utility Functions that
Are Discontinuous at the Origin.
SSRN 2528435: DOI.
We consider the Nash equilibrium existence problem for Cournotian games and we provide two results for it. The first is compatible with utility functions that are discontinuous at the origin, but requires the nonemptiness of best-replies at the origin (and in some sense is known); the second requires the discontinuity of utility functions at the origin and the emptiness of best-replies at the origin (and to the best of our knowledge is not known). Both results are proved by reducing the Nash equilibrium existence problem to a simple fixpoint existence problem for a real function defined on a real interval. The results are then applied to Cournot oligopolies and winner-take-all contest games; various examples illustrate the novelty of the results obtained.
2012 (mit F. Quartieri).
Equilibrium Uniqueness Results for Cournot Oligopolies Revisited.
In: Contributions to Game Theory and Management. Volume V, 209--229.
Editors: L. Petrosjan, N. Zenkevich. Graduate School of Management St. Petersburg.
We revisit and compare equilibrium uniqueness results for
homogeneous Cournot oligopolies. In doing this we provide various useful and
interesting results for which it is difficult to give appropriate reference
in the literature. We also propose problems for future research.
2012 (mit W. Heijman).
New Insights into the Theory of Giffen Goods.
Lecture Notes in Economics and Mathematical Systems, Volume 655. Springer, Berlin.
ISBN 978-3-642-11776-0. Editors: W. Heijman and P. v. Mouche.
One might expect that after their identification in the 19th century,
all aspects of Giffen goods would have been studied by now. This
appears not to be the case. This book contains the latest insights
into the theory of Giffen goods. In the past, surprisingly
few goods could be categorized as ''Giffen''. This may be because
of a lack of understanding the character of these goods.
Therefore, the theories explained in this book may also
produce a solid basis for further empirical research in the field. Experts throughout the world have contributed to this book, which predominantly pursues a mathematically rigorous approach. It may
be used by researchers in the field of fundamental economics and in
graduate-level courses in
2012 (mit W. Heijman).
A Child Garden of Concrete Giffen utility Functions: a Theoretical Review.
In: New Insights into the Theory of Giffen Goods.
Lecture Notes in Economics and Mathematical Systems, 655, 39--52. Springer, Berlin.
Editors: W. Heijman, P. von Mouche.
[ Verbesserung von: 2009 (mit W. Heijman).
On Simple Concrete Giffen Utility Functions: Old and New Results.
Mansholt Graduate School, Discussion Paper 45.
We present a theoretical review of the literature on concrete utility functions for
Giffen and inferior goods within the context of the utility maximisation problem under a budget
restriction and provide new functional forms. The presentation is organised around the specific properties such utility functions
have. These properties include strict increasingness, quasi-concavity, and the applicability of Gossen's second law.
2012 (mit W. Pijnappel).
On the Definition of Giffen and Inferior Goods.
In: New Insights into the Theory of Giffen Goods.
Lecture Notes in Economics and Mathematical Systems, 655, 69--88. Springer, Berlin.
Editors: W. Heijman, P. von Mouche.
We consider Giffen goods for the very general setting of an upper semi-continuous utility function.
Two not equivalent definitions for this notions are discussed. Concerning this we show that there exists a
(well-defined) non-decreasing demand function ℝ++ → ℝ
that is nowhere strictly increasing. Also we reconsider the relation between Giffen and inferior goods.
On Games with Constant Nash Sum.
In: Contributions to Game Theory and Management. Volume IV, 294-310.
Editors: L. Petrosjan, N. Zenkevich.
Graduate School of Management St. Petersburg.
A class of games in strategic form with the following property is identified:
for every n ∈ E, i.e. Nash equilibrium,
the (Nash) sum ∑_l nl is constant. For such a game sufficient conditions for
to be polyhedral and semi-uniqueness (i.e. # E ≤ 1) are given.
The abstract results are illustrated by applying them to
a class of games that covers various types of Cournot oligopoly and
transboundary pollution games.
The way of obtaining the results is by analysing
so-called left and right marginal reductions.
2010 (mit R. Haagsma).
Equilibrium Social Hierarchies: a Non-Cooperative Ordinal Status Game.
The B.E. Journal of Theoretical Economics, Volume 10, Issue 1 (Contributions), Article 24.
We analyze a game in strategic form, where each player′s payoff depends on his action and
his social status, which is given by his rank in the actions distribution. Our focus is on the relation
between the degree of heterogeneity among status-seeking players and the distribution of
their Nash equilibrium actions. We find that if among players intrinsic concerns are sufficiently
important relative to status concerns, individual equilibrium actions diverge, but if status concerns
are relatively important, individual equilibrium actions are the same. Another key result of the
analysis is that, in contrast to what is usually claimed, status seeking need not always be socially
inefficient. If players are sufficiently heterogeneous, there exists a Nash equilibrium that is unique,
separating, and Pareto efficient.
2009. Non-differentiability of Payoff Functions and Non-uniqueness of Nash Equilibria.
World Academy of Science, Engineering and Technology, 53, 731--736.
Given non-degenerate intervals Xi of ℝ and an increasing ordered mapping Φ: X1 x … x XN →
ℝN, games in
strategic form between N players with the Xi as action sets
with the following three properties are studied: the set of Nash equilibria E is convex,
Φ is constant on E
and in each Nash equilibrium at least one payoff function is not partially differentiable
w.r.t. its own action. The results are illustrated for a special class of aggregative games that include
the formal transboundary pollution games with global transboundary pollution.
2008. Direct Sum Games.
Lecture Notes in Management Science, Volume 1, 230--143.
The formal mathematical structure of direct sum games is studied, in particular their relation
with multiobjective games.
A canonical analysis is performed and the notion of equilibrated direct sum game is introduced.
For games in the latter class, and even for not too large deviations of them,
there is special attention for strictly individually rational full cooperative
2007 (mit W. Pijnappel und J. Rouwendal).
On Traffic Congestion Models à la Mohring and Harwitz.
In: Regional Externalities, 71--89, Springer, Berlin. Editor: W. Heijman.
As far as we know, the problem of existence of a maximiser of the
welfare function in traffic congestion models à la Mohring and
Harwitz has never been posed. In this paper we present a mathematically rigorous existence result
and take the opportunity to provide a similar variant of their self-financing result.
2004 (mit H. Folmer).
On a Less Known Nash Equilibrium Uniqueness Result.
Journal of Mathematical Sociology, 28, 67--80.
We generalize a less known Nash equilibrium uniqueness result
for games in strategic form. Its
power is illustrated by applying it to
a Public Goods Game, a Homogeneous Cournot Oligopoly Game
and a Formal Transboundary Pollution Game.
2002 (mit H. Folmer).
The Acid Rain Game: A Formal and Mathematically Rigorous Analysis.
In: Festschrift for K. Mäler, 138--161.
Edward Elgar, Cheltenham. Editors: K. Bengt, P. Dasgupta, K. Lofgrun.
2000 (mit H. Folmer).
Transboundary Pollution and International Cooperation.
In: The International Yearbook of Environmental and Resource Economics 2000/2001,
231-166. Edward Elgar, Cheltenham. Editors: T. Tietenberg, H. Folmer.
1995. Spectral Asymptotics of Periodic Discrete Schrödinger Operators, I.
Asymptotic Analysis 11, 263--187.
Asymptotics, for coupling constant A tending to zero and to infinity,
for the bands and gaps of the Schrödinger operator (HA g)(n) = g(n+1) + Abng(n) + g(n-1)
on l2(ℤ) with periodic potential b are studied. In this context special attention is paid to `asymptotic spectral powers'.
Three different perturbation techniques are used to obtain results. In this first paper
the emphasis lies on results that can be obtained by a first-order degenerate or
arbitrary-order non-degenerate Rayleigh-Schrödinger perturbation calculus.
1994 (mit H. Folmer).
Interconnected Games and International Environmental Problems. II,
Annals of Operations Research, 54, 97--117.
Some features of international environmental problems are considered.
A basic problem is to induce countries to adopt a cooperative approach.
One of the instruments to induce countries to cooperate is an exchange of concessions in fields of relative strengths, such as swapping trade concessions
for cooperation on international environmental problems. This instrument will be modeled in this paper with tensor games.
Both tradeoff and non-tradeoff tensor games will be addressed with emphasis on
tradeoff tensor games with linear strict weights. The relationship between the Pareto-equilibria
of a non-tradeoff tensor game and the Nash-equilibria of the associated tradeoff tensor games
will be studied. Due to structural similarities between tensor games
and repeated multiple objective games, some attention will be also paid to the latter. Relationships between
objects related to Folk theorems for the tradeoff tensor game with completely additive weights
and the corresponding objects for its constituting isolated games will be studied. As many international environmental
problems have prisoners' dilemma characteristics, it is analyzed how interconnection may enhance cooperation in prisoners' dilemma games.
1993 (mit H. Folmer).
Operations Research '92, 10--12, Physica-Verlag, Heidelberg.
Editors: A. Karmann, K. Mosler, M. Schader, G. Uebe.
Dieser Artikel betrifft eine ausfürliche Kurzfassung von
``1994 (mit H. Folmer) Interconnected Games and International Environmental Problems. II,
Annals of Operations Research, 54, 97-117''.
1993 (mit H. Folmer und S. Ragland).
Interconnected Games and International Environmental Problems.
Environmental & Resource Economics 3, 313--335.
[1993 Also in: The Use of Economic Sanctions in Trade and Environmental
Policy, Research Memorandum 9307 Erasmus Universiteit Rotterdam.]
[2015 And also in:
Handbook on Game Theory and International Environmental Cooperation: Essential Readings. Cheltenham, U.K.:
Edward Elgar, 215--237.
(The International Library of Critical Writings in Economics Series) Editors: M. Finus and A. Caparrós.
The purpose of this paper is to introduce the concept of interconnected games
and to show its relevance for modeling international environmental problems. It is argued that
an interconnected game approach to international environmental problems may enhance cooperation and
provide an alternative to the use of financial side payments to induce countries to
cooperate. Two types of interconnected games are distinguished
in this paper, i.e. direct sum games and tensor games. In the former
all the constituting isolated games are games in strategic form and
in the latter they are repeated games. In both cases the interconnected game can be
interpreted as a multiple objective game, but
only the setting where a trade-off is made for the vector-payoffs is considered.
In addition to the formal definition of these types
of interconnected games, some elementary results concerning Nash equilibria of such games are derived.
1992 (mit W. Heijman).
Resource policies, Stability and the Business Cycle.
In: Wageningse Economische Studies 24, 215-136. Editors: W. Heijman and
This essay deals with two extensions of classic business cycle models in the context of
resource economics. The first is based on the traditional post-Keynesian multiplier accelerator
model and is a typical demand side dynamic macroeconomic model. It explains the business cycle for
resource use out of changes in the effective demand. The second, a
dynamic microeconomic model, based on the
traditional cobweb model, explains the variation in the price level of resources. Both models
are described by second order linear difference equations. Special attention is given to
the problem of asymptotic stability in the case of Floquetian coefficients. Furthermore
each model is used to analyse a specific economic policy problem.
1990 (mit J. Avron und B. Simon).
On the Measure of the Spectrum for the Almost Mathieu Operator.
Communications in Mathematical Physics, 132, 103--118.
[1991 Erratum: Communications in Mathematical Physics, 139, 215.]
We obtain partial result on the conjecture that for the almost Mathieu operator
at irrational frequency, α, the measure of the spectrum,
S(α, λ, θ) = | 4 - 2 | λ | |.
For |λ| ≠ 2 we show that if αn is rational and
αn → α irrational, then
S+(αn, λ) → | 4 - 2 | λ| |.
1989. The Coexistence Problem for the Discrete Mathieu Operator.
Communications in Mathematical Physics, 122, 23--33.
We solve the coexistence problem for the periodic discrete Mathieu operator in all parametric cases.
The main tool in the proof will be Bezout's theorem for projective plane curves. As an
additional result we obtain the gap opening and gap growth powers for this operator.
1988. Sur les Régions Interdites du Spectre de
l'Opérateur Périodique et Discret de Mathieu.
Thèse, Rijksuniversiteit Utrecht.
1987. Clustering and Nesting of Energy Spectra.
Proceedings ICIAM87, Paris la Vilette, Mathematisch Centrum Amsterdam, 389-408.
In some models of solid state physics, mysterious phenomena can be observed:
The clustering and nesting of energy spectra. This happens in particular with models leading
to the discrete Mathieu equation
g(n+1) + (2 A cos(2 π n α - ν) - ε) g(n) + g(n-1) =0 (n ∈ ℤ).
Hofstadter (1976) gave an empirical description of the combinatorics of both phenomena.
In this paper we shall give a natural explanation of the combinatorics by introducing
and applying the concept of infinitesimal clustering.
1983. Sur un Electron de Bloch dans un Champ Magnétique et sur l'Equation Presque-Mathieu.
Mémoire, Katholieke Universiteit Nijmegen.
En partant de l'équation de Schrödinger pour un électron qui se
déplace dans un potentiel périodique, soumis en même temps
s à un champ magnétique extérieur, nous arrivons à l'équation
presque-Mathieu, étant une
équation aux différences.
Inspiré par les résultats théoriques pour
l'équation différentielle de Mathieu,
nous obtenons des résultats semblables
pour l'équation presque Mathieu.
En outre nous calculons a l'aide d'un ordinateur puissant
le spectre d'énergie de cette équation.
En particulier nous obtenons des variantes de papillons de Hofstadter.